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Existence and uniqueness of solutions for the generalized Korteweg-de Vries equation. (English) Zbl 0662.35114

We continue the study of the Cauchy problem for the generalized Korteweg- de Vries equation \(\partial_ tu+D^ 3u=DV'(u)\) with \(V\in {\mathcal C}^ 1({\mathbb{R}},{\mathbb{R}})\) and with initial data \(u_ 0\) in \(L^ 2\) or \(H^ 1\) initiated by the first and the second author in a previous paper [[1] Uniqueness of solutions for the generalized Korteweg-de Vries equation, preprint Orsay (1988)]. In [1] uniqueness of the solutions was proved in suitable function spaces and suitable assumptions on V, for \(u_ 0\) in \(L^ 2\) or \(H^ 1\) spaces. Here, using compactness methods, we prove the existence of \(L^ 2\) and \(H^ 1\) solutions for initial data \(u_ 0\) in \(L^ 2\) or \(H^ 1\) under more general assumptions on V, including some restrictions at zero and infinity but no more local regularity than \(V\in {\mathcal C}^ 1\). We then specialize our results to the situations considered in [1] and prove the existence of solutions under the assumptions of [1] in the function spaces considered in [1], thereby converting the uniqueness results of [1] into existence and uniqueness results in the same spaces.
Reviewer: J.Ginibre

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35G25 Initial value problems for nonlinear higher-order PDEs
35B65 Smoothness and regularity of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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References:

[1] Ginibre, J., Tsutsumi, Y.: Uniqueness of solutions for the generalized Korteweg-de Vries equation SIAM J. Math. Anal. (in press) · Zbl 0702.35224
[2] Hartman, P.: Ordinary differential equations. Boston Basel Stuttgart: Birkhäuser 1982 · Zbl 0476.34002
[3] Kato, T.: On the Cauchy problem for the generalized Korteweg-de Vries equation. In: V. Guillemin (ed.) Studies in Applied Mathematics. Adv. Math., Suppl. Stud.18, 93–128 (1983) · Zbl 0549.34001
[4] Kruzhkov, S.N., Faminskii, A.V.: Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation. Math. USSR, Sb.48, 391–421 (1984) · Zbl 0549.35104 · doi:10.1070/SM1984v048n02ABEH002682
[5] Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod and Gauthier Villars, 1969
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